Theorem frgrncvvdeqlem1 30503

Description: Lemma 1 for frgrncvvdeq 30513. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e	⊢ 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny	⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem1	⊢ (𝜑 → 𝑋 ∉ 𝑁)

Proof of Theorem frgrncvvdeqlem1

Step	Hyp	Ref	Expression
1		frgrncvvdeq.xy	. . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
2		df-nel 3064	. . . . 5 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷)
3		frgrncvvdeq.nx	. . . . . 6 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4	3	eleq2i 2856	. . . . 5 ⊢ (𝑌 ∈ 𝐷 ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
5	2, 4	xchbinx 336	. . . 4 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
6	1, 5	sylib 220	. . 3 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7		nbgrsym 29566	. . 3 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
8	6, 7	sylnibr 331	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9		frgrncvvdeq.ny	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
10		neleq2 3070	. . . 4 ⊢ (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
11	9, 10	ax-mp 5	. . 3 ⊢ (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12		df-nel 3064	. . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
13	11, 12	bitri 277	. 2 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
14	8, 13	sylibr 236	1 ⊢ (𝜑 → 𝑋 ∉ 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   ∉ wnel 3063  {cpr 4586   ↦ cmpt 5183  ‘cfv 6523  ℩crio 7354  (class class class)co 7398  Vtxcvtx 29199  Edgcedg 29250   NeighbVtx cnbgr 29535   FriendGraph cfrgr 30462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-nbgr 29536

This theorem is referenced by:  frgrncvvdeqlem7  30509  frgrncvvdeqlem8  30510  frgrncvvdeqlem9  30511